Some results related to group actions in several complex variables

TL;DR

This paper discusses recent advances in group actions in several complex variables, highlighting differences between compact and noncompact groups, including solutions to longstanding conjectures and properties of automorphism groups.

Contribution

The paper presents new results on the extended future tube conjecture and invariant extension of holomorphic functions, distinguishing compact and noncompact group actions.

Findings

Extended future tube is a domain of holomorphy.

Automorphism groups of invariant domains can be compact.

Invariant extension of holomorphic functions is established.

Abstract

In this talk, we'll present some recent results related to group actions in several complex variables. We'll not aim at giving a complete survey about the topic but giving some our own results and related ones. We'll divide the results into two cases: compact and noncompact transformation groups. We emphasize some essential differences between the two cases. In the compact case, we'll mention some results about schlichtness of envelopes of holomorphy and compactness of automorphism groups of some invariant domains. In the noncompact case, we'll present our solution of the longstanding problem -- the so-called extended future tube conjecture which asserts that the extended future tube is a domain of holomorphy. Invariant version of Cartan's lemma about extension of holomorphic functions from the subvarities in the sense of group actions will be also mentioned.